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Darrell Jenkins
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1 HEIGHTS AND DISTANCES 1. If te angle of elevation of cloud from a point meters aove a lake is and te angle of depression of its reflection in te lake is cloud is., prove tat te eigt of te Ans : If te angle of elevation of cloud from a point n meters aove a lake is and te angle of depression of its reflection in te lake is β, prove tat te eigt of te tan β + tanα cloud is tan β tanα Let AB e te surface of te lake and Let p e an point of oservation suc tat AP meters. Let c e te position of te cloud and c e its reflection in te lake. Ten CPM and MPC 1 β. Let CM x. Ten, CB CM + MB CM + PA x + CM In CPM, we ave tan PM x tan AB [ PM AB] AB x cot In PMC, we ave C' M tanβ PM..1 x + 2 tanβ [Θ C MC B+BM x + + n] AB AB (x + 2) cot β From 1 & 2 x cot (x + 2) cot β 49

2 x (cot - cot β) 2 cot β (on equating te values of AB) tan β tanα 2 x x tanα tan β tan β tanα + tan β tan β 2 tanα x tan β tanα Hence, te eigt CB of te cloud is given y CB is given y CB x + 2 tanα CB + tan β tanα 2 tanα + tan β tanα (tanα + tan β ) CB- tan β tanα tan β tanα 2. From an aero plane vertically aove a straigt orizontal road, te angles of depression of two consecutive milestones on opposite sides of te aero plane are oserved to e α and β. Sow tat te eigt of te aero plane aove te road is. Let P Q e QB e x Given : AB 1 mile QB x AQ (1- x) mile in PAQ PQ Tan α AQ Tan α 1 x 1 x Tanα.1 In PQB Tan β x x Tanβ Sustitute for x in equation (1) 1 + Tanβ Tanα 50

3 Tanβ Tanα 1 Tan β + Tanα TanβTanα. Two stations due sout of a tower, wic leans towards nort are at distances a and from its foot. If α and β e te elevations of te top of te tower from te situation, prove tat its inclination θ to te orizontal given y Let AB e te leaning tower and C and D e te given stations. Draw BL DA produced. Ten, BAL 0, BCA α, BDC a and DA. Let AL x and BL In rigt ALB, we ave : AL x Cot θ Cot θ BL x Cot θ x cot θ..(i) In rigt BCL, we ave : CL Cot α a + x cot α BL a (cot α - cot θ) a (cotα cotθ )...(ii) In rigt BDL, we ave : DL DA + cot β cot β BL BLAL + x cot β + x cot β ((cot β - cot θ) (cot β cotθ ) [using (i)]..(iii) equating te value of in (ii) and (iii), we get: a (cotα cotθ ) (cot β cotθ ) 51

4 a cot β - a cot θ cot α - cot θ ( a) cot θ cot α - a cotβ cotα a cot βθ cot θ ( a) 4. Te angle of elevation of te top of a tower from a point on te same level as te foot of te tower is α. On advancing p meters towards te foot of te tower, te angle of elevation ecomes β. sow tat te eigt of te tower is given y 5. A oy standing on a orizontal plane finds a ird flying at a distance of 100m from im at an elevation of 0 0. A girl standing on te roof of 20 meter ig uilding finds te angle of elevation of te same ird to e Bot te oy and te girl are on opposite sides of te ird. Find te distance of te ird from te girl. ( 42.42m) In rigt ACB AC Sin 0 AB 1 AC AC 100 AC 50m AF (50 20) 0m In rigt AFE AF Sin 45 AE AE AE x m 6. From a window x meters ig aove te ground in a street, te angles of elevation and depression of te top and te foot of te oter ouse on te opposite side of te street are α and β respectively. Sow tat te eigt of te opposite ouse is Meters. Let AB e te ouse and P e te window Let BQ x PC x Let AC a 52

5 PQ In PQB, tan θ or tan θ QB x x cot θ tanθ AC a In PAC, tan θ or tan θ PC x a x tan θ > ( cot θ) tan θ tan θ cot θ. te eigt of te tower AB AC + BC a + tan θ cot θ + (tanθ cot θ + 1) 7. Two sips are sailing in te sea on eiter side of a ligtouse; te angles of depression of two sips as oserved from te top of te ligtouse are 60 0 and 45 0 respectively. If te distance etween te sips is meters, find te eigt of te ligtouse. (200m) In rigt ABC Tan 60 BC BC H BC In rigt ABD Tan 45 BD BD BC + BD 200 BC + BC 200 BC 200( ( 1+ ) 1+ BC m eigt of ligt ouse 200m 5

8. A round alloon of radius a sutends an angle θ at te eye of te oserver wile te angle of elevation of its centre is Φ. Prove tat te eigt of te center of te alloon is a sin θ cosec Φ /2.

6 8. A round alloon of radius a sutends an angle θ at te eye of te oserver wile te angle of elevation of its centre is Φ. Prove tat te eigt of te center of te alloon is a sin θ cosec Φ /2. Let θ e te centre of te allon of radius r and p te eye of te oserver. Let PA, PB e tangents from P to allong. Ten APB θ. APO BPO 2 θ Let OL e perpendicular from O on te orizontal PX. We are given tat te angle of te elevation of te centre of te allon is φ i.e., OPL φ θ OA In OAP, we ave sin 2 OP θ a sin 2 OP θ OP a cosec 2 OL In OP L, we ave sinφ OP OL OP sin φ a cosec 2 φ sin θ. Hence, te eigt of te center of te alloon is a sin θ cosec Φ /2. 9. Te angle of elevation of a jet figter from a point A on te ground is After a fligt of 15 seconds, te angle of elevation canges to 0 0. If te jet is flying at a speed of 720 km/r, find te constant eigt at wic te jet is flying.(use 1.72 ( 2598m) 6 km / r 10m / sec 720 km / 10 x Speed 200 m/s Distance of jet from AE speed x time 200 x m tan 60 o AC oppositeside BC adjacentside AC BC BC AC 54

7 AC ED (constant eigt) BC ED.1 tan 0 o ED oppositeside BC + CD adjacentside 1 ED BC BC ED BC BC (from equation 1) BC BC BC BC BC 000 BC BC 1500 m ED BC (from equation 1) x 1.72 ED 2598m Te eigt of te jet figter is 2598m. 10. A vertical post stands on a orizontal plane. Te angle of elevation of te top is 60 o and tat of a point x metre e te eigt of te post, ten prove tat Self Practice 2 x. 11. A fire in a uilding B is reported on telepone to two fire stations P and Q, 10km apart from eac oter on a straigt road. P oserves tat te fire is at an angle of 60 o to te road and Q oserves tat it is an angle of 45 o to te road. Wic station sould send its team and ow muc will tis team ave to travel? (7.2km) Self Practice 12. A ladder sets against a wall at an angle α to te orizontal. If te foot is pulled away from te wall troug a distance of a, so tat is slides a distance down te wall cosα cos β a making an angle β wit te orizontal. Sow tat. sin β sinα Let CB x m. Lengt of ladder remains same 55

8 CB Cos α CA x Cos α x cos α ED AC Let Ed e ED AC (1) DC + CB cos β ED a + x cos β a + x cos β x cos β a (2) from (1) & (2) cos α cos β - a cos α - cos β - a -a (cosα - cosβ)...() AE + EB Sin α AC + EB Sin α Sin α EB EB Sin α...(4) EB Sin β DE EB Sin β EB Sin β From (4) & (5) (5) Sin β Sin α sin α Sin β - (Sin β - Sin α)...(6) Divide equation () wit equation (6) a (cosα cos β ) (sin β Sinα) a Cosα Cosβ Sinβ Sinα 56

9 1. Two stations due sout of a leaning tower wic leans towards te nort are at distances a and from its foot. If α, β e te elevations of te top of te tower cotα acot β from tese stations, prove tat its inclination ϕ is given y cotϕ. a Let AE x, BE BE Tan φ AE x 1 x x tanφ x cot φ BE tan α CE a + x a + x cot α x cot α - a BE tan β DE + x +x cot β x cot β from 1 and 2 cot φ cot α - a ( cot φ + cot α ) a a cot φ + cotα from 1 and cot φ cot β - ( cot φ - cot β) cot φ + cot β from 4 and 5 a cot φ + cotα cot φ + cot β a (cot β - cot φ ) ( cot α - cot φ ) - a cot φ + cot φ cot α - a cot β ( a) cot φ cot α - a cot β 57

10 cot α - a cot β cot φ a 14. In Figure, wat are te angles of depression from te oserving positions O 1 and O 2 of te oject at A? Self Practice ( 0 o,45 o ) 15. Te angle of elevation of te top of a tower standing on a orizontal plane from a point A is α. After walking a distance d towards te foot of te tower te angle d of elevation is found to e β. Find te eigt of te tower. ( ) cotα cot β Let BC x AB tan β CB tan β x x tan β x cot β (1) AB tan α DC + CB tan α d + x d + x cotα tan α x cot α - d (2) from 1 and 2 cot β cot α - d (cot α - cot β ) d d cot α cot β 58

11 16. A man on a top of a tower oserves a truck at an angle of depression α were 1 tanα and sees tat it is moving towards te ase of te tower. Ten minutes 5 later, te angle of depression of te truck is found to e β were tan β 5, if te truck is moving at a uniform speed, determine ow muc more time it will take to reac te ase of te tower minutes600sec A Let te speed of te truck e x m/sec CDBC-BD In rigt triangle ABC tanα BC 1 tanα 5 BC 5. 1 In rigt triangle ABD tanβ BD C α D β B 5BD ( tan β 5 ) CDBC-BD ( CD600x ) 600x 5BD-BD BD150x 150x Time taken x 150 seconds Time taken y te truck to reac te tower is 150 sec. 59

Downloaded from APPLICATION OF TRIGONOMETRY

Downloaded from APPLICATION OF TRIGONOMETRY MULTIPLE CHOICE QUESTIONS APPLICATION OF TRIGONOMETRY Write the correct answer for each of the following : 1. Write the altitude of the sun is at 60 o, then the height of the vertical tower that will cost